mathematical concept which does not have meaning and so which is not assigned an interpretation
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What is an example of an undefined expression?
Is 0 over something undefined?
What does it mean when expression is undefined?
The problem is that undefined compared to null using == gives true. The common check for undefined is therefore done like this:
typeof x == "undefined"
this ensures the type of the variable is really undefined.
It turns out that you can set window.undefined to whatever you want, and so get object.x !== undefined when object.x is the real undefined. In my case I inadvertently set undefined to null.
The easiest way to see this happen is:
window.undefined = null;
alert(window.xyzw === undefined); // shows false
Of course, this is not likely to happen. In my case the bug was a little more subtle, and was equivalent to the following scenario.
var n = window.someName; // someName expected to be set but is actually undefined
window[n]=null; // I thought I was clearing the old value but was actually changing window.undefined to null
alert(window.xyzw === undefined); // shows false
The "first level" answer should be: "undefined" plainly means "not defined", i.e. the expression has no value. For example, $\frac{a}{0}$ is undefined for all $a\in\mathbb R$, $\tan(k\pi+\pi/2)$ is undefined for all $k\in\mathbb Z$, $\log(x)$ is undefined for $x\le 0$. This means: those expressions have no meaning, don't write them, do not divide by zero, don't calculate tangent of an odd multiple of $\pi/2$ or a logarithm of a negative number (or zero).
The "second level" answer should be that people have tried to assign some value to some of these expressions (i.e. it's not that we were lazy!), but have found that there is no universal answer. There are partial answers, depending on the context. Thus, the consensus is: at the "first level" say that those expressions are undefined, and then, at the second level, do define them in various contexts, but be aware of the limitations. This is an instance of "walk before you run".
This is also a story of compromise: you gain something (by defining something) but you also lose something (by sacrificing some of things that you are taking for granted).
For example, the infinity. Why don't we "just add" another number, call it "infinity", label it with $\infty$, and define $\frac{1}{0}:=\infty$? What would go wrong? As it happens, a lot. $\mathbb R$ is not just a set, it is a field, i.e. a very regular structure with addition, subtraction, multiplication and division. How do those extend to $\infty$? Is $\frac{2}{0}=\infty$ too? Is $\infty=\frac{1}{0}=\frac{2-1}{0}=\frac{2}{0}-\frac{1}{0}=\infty-\infty=0$? This "paradox" shows you that, whatever you decide $\frac{1}{0}$ to be, you cannot expect the ordinary arithmetic rules to stay valid. So you have to sacrifice something: if it is not the ability to calculate $\frac{1}{0}$, it is the ability to apply the laws of arithmetic!
The latter sacrifice seems bigger, but it isn't always, and it isn't in all contexts. We must still say "infinity is not a number" to remind ourselves to not use arithmetic operations on it, but we can extend the topology of $\mathbb R$ ("compactify it with one point") and talk about convergence. This is what you will be doing in Calculus. Except - as it happens, there is another, equally valid, and complementary, way to extend $\mathbb R$ with infinities: don't add one infinity $\infty$ but add two infinities: $+\infty$ and $-\infty$. (Again, don't do any arithmetic on them!)
Sometimes you can extend "undefined" expressions without any hassle: $\frac{\sin x}{x}$ is undefined for $x=0$, but not only that $\lim_{x\to 0}\frac{\sin x}{x}=1$ is well-defined; it is, in a way, a part of the function. Namely, when you "patch up" $\frac{\sin x}{x}$ to "become" $1$ for $x=0$, you are not patching up anything - you are discovering a new reality. The function "patched up" in such way is a very nicely behaved, in fact it is an analytic function on the whole $\mathbb R$ (and even on the whole $\mathbb C$). On the other hand, if you try to "patch up" $\frac{1}{x}$ at $x=0$, whatever you do you cannot get even a continuous function.
Somewhere "in-between" is the case of the logarithm. Expand $\mathbb R$ into $\mathbb C$, and suddenly you have a logarithm (except for $x=0$ - this one stays undefined) - but you get more than you've bargained for: you can solve $e^y=x$ for any $x\ne 0$ but $y$ is not unique. Thus, people usually restrict the range of $y$'s to a horizontal strip of width $2\pi$ in the complex plane. So again: you gain (can define logarithm) but you lose (which strip you've chosen is arbitrary, and some rules, e.g $\log(ab)=\log a+\log b$, aren't valid anymore: instead, $\log(ab)\equiv\log a+\log b\pmod{2\pi i}$).
Division is defined as the inverse of multiplication. Multiplication by zero is defined as always giving the answer zero. Given $b\times0=0$ there is not any number which you can multiply the right hand $0$ by and get $b$ back.
If we say that there is such a number $0^{-1}$ such that $0\times 0^{-1}=1$ then what about $2\times0=0$? What would be the inverse of that such that $0\times 0^{-1}=2$ as well as $0\times 0^{-1}=1$?
Suppose I define a function f with the domain $[0,1]$, then $f(-1)$ is undefined, just like $f(\text{banana})$ is undefined, and in exactly the same way $1/0$ is not defined. The tangent example is not defined because division by zero is not defined. It really has absolutely nothing to do with anything approaching infinity.
The resolution of the paradox "what happens when an unstoppable force meets an immovable object" is that you cannot have an unstoppable force in the same universe as an immovable object. The resolution of zero times everything equals zero is that you cannot define an inverse of multiplication by zero.
Saying that 1 divided by 0 is undefined, does not mean that you can carry out the division and that the result is some strange entity with the property “undefined”, but simply that dividing 1 by 0 has no defined meaning. That is just like when you ask whether the number 1.9 is odd or even: That is not defined. Or when you ask what colour the number 7 has.
To put matters straight: Division is a function $$q:\quad{\mathbb R}\times{\mathbb R}^*, \qquad(a,b)\mapsto q(a,b)=:{a\over b}\ ,$$ whereby $q(a,b)$ is the unique number $x\in{\mathbb R}$ such that $b \>x=a$.
When we say that $\displaystyle{a\over0}$ is undefined then this means no more and no less than that the pair $(a,0)$ is not in the domain of the function $q(\cdot,\cdot)$.
Now to your three ways of understanding "undefined" in the realm of division by $0$:
If $\displaystyle{a\over0}$ could be any number, say $=13$, then this would enforce $13\cdot0=a$, which is wrong when $a\ne0$.
This is even worse. Why should $\displaystyle{7\over0}$ be the Eiffel tower?
There are circumstances where division by zero makes sense, e.g. in connection with maps of the Riemann sphere, or with meromorphic functions. There one has $\infty$ as an additional point in the universe of discourse. But these circumstances require special exception handling measures, and the "usual rules of algebra" are not valid when dealing with $\infty$.